The Power of Practice

Maths “ability” is just practice. For all the rhetoric we hear about ability in maths, any innate component is of relatively little consequence. What appears to be ability is just practice masquerading as talent. But despite this we have a Western obsession with innate ability. Culturally we treat Maths as something you are either born good at, or born bad at, and those in the latter camp should brashly accept it. After all, if your genes make you bad at something why should you be shy about it?

But innate ability is a distraction from the real cause of good performance – practice. Practice is the single most important component of success in Maths.

A. Connections strengthen memories. As we learn more, and so have more associations to make with each new piece of content, we remember more. These new connections in turn strengthen our old knowledge, and contribute to the wider structure of knowledge that forms our view of the world. This often looks like innate ability. A student who knows about fractions will learn more from a lesson on percentages than a student who doesn’t know about fractions. We might hypothesise that this is because they are “better at Maths”. In fact it’s because they can build connections with their fractions knowledge, and so understand percentages better (and are more able to remember this new knowledge).

B. Mastery frees up working memory. Our working memories have a limited capacity. Once they’re full they’re full, and we cannot process any more information. One model of working memory conceptualises it as having a fixed number of slots. Each slot can hold one piece of information, like a step in a method or a number being remembered. Trying to carry out a complex process can easily overload working memory as all the slots get filled and thinking grinds to a halt – meaning the thinker is unable to learn. However fluency bypasses the constraints of working memory. Once you have mastered a procedure you carry it out without thinking – it does not occupy multiple slots in your working memory but frees it up to learn instead. This means that the student who has practised to the point of fluency will learn faster, and have an appearance of greater ability.

C. Fluency solves problems. Because fluency frees up working memory, it gives students the space to think about challenging problems and come up with creative solutions. The most well-practised students will therefore come up with ideas that challenge their thinking, and so learn more at the boundaries of their knowledge.

Getting better at practice must be a national priority. We have a national fear of practice. When being trained as a Maths teacher I was told to “Beware the three Xs. Examples, exercises, and more exercises.” The intention – that incredibly dull lessons should be avoided – was a good one. But there is no need to make practice taboo. It is the hallmark of elite performance, a necessary prerequisite to any worthwhile achievement. It is not an end in itself, but the ends we seek cannot be accomplished without significant practice.

There are two components of this challenge to practice better. We need to:

1. Increase the quantity of practice.
Every time a new international study shows that our relative performance is worse than the Far East there is an outcry from people who argue that the outcomes are just not comparable. It is not fair, they argue, to compare UK results to the results of Singapore, Shanghai or South Korea, because in those countries children do hours of homework, have private tutors, and spend far longer practising. This is not an excuse. If the quantity of practice is the cause for this gap in performance then we need to get that same quantity of practice for our children.

2. Increase the quality of practice.
Good practice means learning the right thing, the right way, and with the right feedback. For this we need:

Better resources – well thought out sets of practice questions that develop understanding and challenge thinking. No current UK textbooks (that I have found!) are up to scratch, and downloads from TES certainly aren’t. Schools and groups of schools need to put real effort into designing the best resources so that teachers can stop reinventing the wheel and focus on how to teach to the specific group of students in front of them.

Better teaching – teachers need deep subject expertise to make sure students receive good quality explanations and are guided through the practice properly. Subject specialism is important here, but so is ongoing professional development. No specialist is too good to carry on learning, so without a better culture of effective CPD we will continually miss out on the potential of the workforce.

Better feedback – good feedback is both precise and timely. It must be precise enough to guide you to action (no vague targets please), and timely enough for you to correct any misconceptions before they settle. Without precise and timely feedback your practice could make you learn mistakes rather than the actual concept itself.

The hallmark of elite performance is a relentless dedication to practice. Sportspeople drill basic techniques over and over again, making them instinctive so that in the heat of a game they can be deployed effortlessly. Musicians practise pieces for days and weeks so that they can begin to play them with feeling and creativity. So too must students practise basic techniques until they are fluent, before they can enjoy the dizzying heights of elite performance. Only with fluency can they begin to tackle truly rewarding problems.

Tim Oates describes how, in the drafting stages of the new National Curriculum, he went through battles to have the word “practice” included in the Mathematics section. It was simply too unpalatable. We need to debunk the two pernicious myths: that maths ability is innate, or that fluency is not a prerequisite for good performance. Practice has the power to make everyone good at Maths, and we need to start to unleash it.

This post is based on a presentation I gave to schools at the first Maths Hub Forum in Manchester last month.

It’s far more complex than that. There’s clearly no “Maths gene” that makes you better at Maths. Intelligence is a complex system, in which genes do play a part, but practice itself affects how they are expressed and how they work together in this system. We know, for example, that environmental interventions can increase working memory capacity up to the level needed to be able to master school-level maths. If this is the case then the starting point matters little if we intervene quickly.

I think it’s unhelpful to set up a false dichotomy between drilling short exercises and not doing practise. There is a well-developed tradition of projects or open questions which hide substantial practise (e.g. Mike Ollerton, nrich or Points of Departure).

These have massive advantages over exercises. Unlike exercises, they reflect the nature of maths as an academic discipline. Your analogy with sport is false: academic mathematics doesn’t benefit from supreme fluency in arithmetic or calculus. They suck you in, and motivate a wider proportion of your students. They can lead to in directions you did not expect, and stop you, as teacher, from getting bored by teaching surds the twentieth time!

Continuing the dominance of individual work on short textbook exercises is not the only way of interpreting recent theory on working memory.

A superb post. These things just must be said if more children are to become proficient in maths. After giving my eldest daughter practice to fluency in maths her progress was startling.
Those that say the same progress as is achieved in the Far East can be achieved in other ways need to demonstrate that. Their theories currently dominate maths education here and don’t seem to work very well!
I don’t choose to spend as much time on maths with my children as is given to it in the Far East but the time we do spend involves practice with the aim of fluency because this works.

2. Increase the quality of practice.
Good practice means learning the right thing, the right way, and with the right feedback. For this we need: I was told that I was going to be bad at math based on genetics. if I wanted to start right now and begin a math study. what would you recommend that I practice to learn the right thing the right way and with the right feedback. Text books grade levels , practice methods, etc….